As the Laplacian \(L_n\)  is a symmetric matrix, its right and left eigenvectors do not differ [1]}, and so the relation \(\mathbf {v}_F^T L_n = \lambda _2 \mathbf {v}_F^T\)  follows from equation (REF ). Also, it is to be remembered that the Fiedler vector is normalized, i.e. \(\mathbf {v}_F^T \mathbf {v}_F =1\) , and so equation (REF ) leads to the following expression of the differential of the algebraic connectivity.
\(\text{d}\lambda _2 & = & \mathbf {v}_F^T ~ \text{d}L_n ~ \mathbf {v}_F~.\) 
